Definitely list it - it shows you were working and gaining experience, and gaps in your resume are arguably worse than bad companies unless you have a very good reason for them. Interviewers will almost certainly ask about any gaps on your resume, which means you'll end up telling them about the company anyway, it just makes it look like you were trying to hide the information from them. That's definitely not a good look!
Instead, list it, and when they ask about it and/or why you left, be prepared to discuss what happened there. Try to remain professional when doing so - a little honesty here will get you further than trying to hide the previous role, and additionally it will provide some context when you explain why you're not likely to receive a reference from the company.
In terms of references, try to line up two or three really good character references - particularly people who may have had experience of seeing your work output (perhaps a colleague at your last job if there was anyone there you got on well with, or failing that, perhaps a teacher, or anyone you may have done any voluntary work for). It's not an ideal situation to have a single job on your resume with no reference, but it's also not a unique situation. It happens, and I'm assuming your next role will be relatively junior, and so that kind of situation is expected.